A hybrid decision making method based on  q -rung orthopair fuzzy soft information

1 Introduction

In our daily life problem, we face uncertainties in making right decisions. Several decision making methods have been investigated in medical field. Decision making is a technique of nominating the best option among various objects. That’s why decision making performs an important role in the human life problems. The life style can be changed by a good decision. First of all, a decision maker tries to know about the restrictions, advantages and qualities of each alternative and then he/she finds out the final decision. To handle with such situations, Zadeh [45] proposed the prominent model of fuzzy set (FS). FS brought the revolution in different fields of science. In FS, a membership degree (MD) within the interval [0, 1] is assigned to each object. Atanassov [9] proposed the intuitionistic fuzzy set (IFS) which is characterized by both the MD μ and the nonmembership degree (NMD) ν within the interval [0, 1], respectively such that the summation of these two values is less than or equal to 1. From last few years, different researchers developed various operators on the existing ideas in which the best one idea is the AOs. Yager [40] introduced the dominant concept of ordered weighted (OW) AOs. The concepts of novel geometric AOs for IFS in multi attribute decision making (MADM) were presented by Xu and Yager [39]. The study of intuitionistic fuzzy (IF) AOs was given by Xu [37]. Wang and Liu [36] discussed the IF Einstein weighted geometric (IFEWG) operators. Li [21] studied the generalized OW averaging operators for IF data. Garg and Rani [13] initiated the idea of complex IF AOs and applied this concept in MADM problems. IFS is a robust concept and many researchers have done work on it. However, there are some drawbacks in the model of IFS, due to which this theory fails to handle the problems.

To overcome such difficulties, Yager [42] discussed the model of Pythagorean fuzzy set (PFS). This model has the quality to replace the constraint of IFS with the condition μ² + ν² ≤ 1. Yager [41] proposed PF arithmetic and geometric operators. Peng and Yang [28] developed interval-valued PF AOs. A hybrid structure for PF MADM was proposed by Zeng et al. [46]. Akram et al. [1] worked for the development of Pythagorean Dombi fuzzy AOs with applications. Shahzadi et al. [31] developed the theory of Yager AOs under PF data. However, the PFS has also some limitations, if MD of an element is 0.9 and NMD is 0.6, then sum of square of these values is greater than 1 and PFS cannot deal such situations. That’s why researchers needed some new extensive model for such type of shortcomings.

Yager [43] investigated new notion of q-rung orthopair fuzzy set (q-ROFS) in which μ ^q  + ν ^q  ≤ 1. In decision making problems, q-ROFS is more capable than IFS and PFS, as shown in Fig. 1. All IF numbers (IFNs) and PF numbers (PFNs) are q-rung orthopair fuzzy numbers (q-ROFNs) but converse is not true. Jana et al. [17] studied q-rung orthopair fuzzy Dombi AOs. Joshi and Gegov [20] developed the confidence levels q-rung orthopair fuzzy AOs. Akram and Shahzadi [4] introduced Yager AOs for q-ROFS. Garg [10] defined some trigonometric operations laws for q-ROFS. The concept of neutrality AOs of q-ROFS was studied by Garg and Chen [12]. As FS, IFS, PFS, q-ROFS and so forth are generally applied by researchers to deal with vagueness and uncertainty during analysis but all these concepts have no information about parameterized study.

OPEN IN VIEWER

Fig. 1

Comparison of q-ROFN, PFN and IFN spaces.

In 1999, a pioneer model of S _f S was proposed by Molodstov [27] which maintains the parametrization tool to handle with vague data. Many ideas are combined with S _f S and established new theories. Maji [25, 26] discussed the models of fuzzy S _f S (FS _f S) and intuitionistic fuzzy S _f S (IFS _f S). The intuitionistic fuzzy soft (IFS _f ) AOs were studied by Arora and Garg [8]. Garg and Arora [11] discussed the generalized IFS _f power AOs and their application in MADM. q-rung orthopair fuzzy soft (q - ROFS _f ) average operators were initiated by Hussain et al. [16]. Zhang et al. [47] developed the theory of consensus reaching for social network group decision making. Zhang et al. [48] discussed the concept of consistency improvement for fuzzy preference relations with self-confidence. For other terminologies not discussed in the paper, the readers are suggested to [3, 44].

The motivations of this article are described as:

Advantages of q - ROFS _f S: q - ROFS _f S, a remarkable extension of IFS _f S and Pythagorean fuzzy S_fS (PFS_fS), permits modeling of the situations with higher more generality than IFS _f S and PFS _f S, because it still applies in cases where μ + ν ≤ 1 and μ² + ν² ≤ 1 but it satisfies μ ^q  + ν ^q  ≤ 1 corresponding to different parameters.

The contributions of this article are outlined as:

The idea of Yager AOs is extended to q - ROFS _f numbers (q - ROFS _f Ns) and fundamental properties are discussed.

The remaining paper is composed as follows: In Section 2, some basic definitions are discussed. In Section 3, we study Yager operations for q - ROFS _f Ns, q - ROFS _f YWA operator and discuss some results in the light of these operations. Section 4 presents q - ROFS _f YOWA operator and some properties related to them. In Sections 5 and 6, we discuss q - ROFS _f YWG and q - ROFS _f YOWG operators to aggregate the q - ROFS _f Ns. Section 7 gives an algorithm for MAGDM and discuss a numerical example in the field of medical diagnosis based on q - ROFS _f Ns. Section 8 gives the comparison analysis with existing operators to show the superiority and validity of proposed theory. In Section 9, we have concluded results about proposed theory.

In this section, we recall some basic definitions.

Definition 2.1. [29] Let U be a universe of discourse and W ˜ be the set of all parameters, E ˜ ⊆ W ˜ . P ( U ) represents the set of all PFSs over U . ( J ˜ , E ˜ ) is called a PFS _f S over U , where J ˜ is a function given by J ˜ : E ˜ → P ( U ) , which is J ˜ ( e p ) = { ≺ x t , f p ( x t ) , g p ( x t ) ≻ } | x t ∈ U } , with condition 0 ≤ ( f p ( x t ) ) 2 + ( g p ( x t ) ) 2 ≤ 1 .

Definition 2.2. [16] Let U be a universe of discourse and W ˜ be the set of all parameters, E ˜ ⊆ W ˜ . P ( U ) represents the set of all q-ROFSs over U . ( τ , E ˜ ) is called a q - ROFS _f S over U , where τ is a function given by τ : E ˜ → P ( U ) , which is τ ( e p ) = { ≺ x t , f p ( x t ) , g p ( x t ) ≻ } | x t ∈ U } , with condition 0 ≤ ( f p ( x t ) ) q + ( g p ( x t ) ) q ≤ 1 . π τ e t p = 1 - ( ( f t p ) q + ( g t p ) q ) q is the hesitancy degree, where f t p denotes f p ( x t ) and g t p denotes g p ( x t ) .

Definition 2.3. [41] For any two real numbers x and y, Yager’s t-norms and Yager’s t-conorms are given as: T ( x , y ) = 1 - min ( 1 , ( ( 1 - x ) ϑ + ( 1 - y ) ϑ ) 1 ϑ ) ,(1) T ′ ( x , y ) = min ( 1 , ( x ϑ + y ϑ ) 1 ϑ ) , ϑ ∈ ( 0 , ∞ ) .(2)

In this section, we develop Yager t-norm and t-conorm for q - ROFS _f Ns and Yager weighted average operators for q - ROFS _f Ns.

Definition 3.1. Let τ e 1 p = ( f 1 p , g 1 p ) ( p = 1 , 2 ) , and τ = (f, g) be three q - ROFS _f Ns and λ > 0, ϑ ∈ (0, ∞). Then, Yager norm operations for q - ROFS _f Ns are

1 . τ e 11 ⊕ τ e 12 = ( min ( 1 , ( f 11 q ϑ + f 12 q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( ( 1 - g 11 q ) ϑ + ( 1 - g 12 q ) ϑ ) 1 ϑ ) q ) , 2 . τ e 11 ⊗ τ e 12 = ( 1 - min ( 1 , ( ( 1 - f 11 q ) ϑ + ( 1 - f 12 q ) ϑ ) 1 ϑ ) q , min ( 1 , ( g 11 q ϑ + g 12 q ϑ ) 1 ϑ ) q ) , 3 . λ τ = ( min ( 1 , ( λ f q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( λ ( 1 - g q ) ϑ ) 1 ϑ ) q ) , 4 . τ λ = ( 1 - min ( 1 , ( λ ( 1 - f q ) ϑ ) 1 ϑ ) q , min ( 1 , ( λ g q ϑ ) 1 ϑ ) q ) .

Definition 3.2. The score function and accuracy function for q - ROFS _f Ns τ e t p = ( f t p , g t p ) are represented by S ( τ e t p ) = f t p q - g t p q , S ( τ e t p ) ∈ [ - 1 , 1 ] and H ( τ e t p ) = f t p q + g t p q , H ( τ e t p ) ∈ [ 0 , 1 ] respectively.

Definition 3.3. Let τ _{e 11}  = (f₁₁, g₁₁) and τ _{e 12}  = (f₁₂, g₁₂) be two q - ROFS _f Ns,

if S (τ _{e 11} ) > S (τ _{e 12} ), then τ _{e 11} ≻ τ _{e 12}  ;

Definition 3.4. Let τ e t p = ( f t p , g t p ) ( t = 1 , 2 , ⋯ , n ; p = 1 , 2 , ⋯ , m ) be a collection of q - ROFS _f Ns and w t , v p are the weight vectors (WVs) for the experts x t ′ s and parameters e p ′ s , respectively, satisfying w t > 0 , v p > 0 with ∑ t = 1 n w t = 1 and ∑ p = 1 m v t = 1 . Then q - ROFS _f YWA operator is a function q - ROFS _f YWA: M n → M , where M is the set of q - ROFS _f Ns. q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) = ⊕ p = 1 m v p ( ⊕ t = 1 n ( w t τ e t p ) ) .

Theorem 3.1. The aggregated value by utilizing q - ROFS _f YWA operator is given by q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) = ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) q ) .(3)

Proof. Applying mathematical induction to prove it.

(i) When n = 2 , m = 2 , q - ROFS _f YWA (τ _{e 11} , τ _{e 12} ) = ⊕ p = 1 2 v p ( ⊕ t = 1 2 w t τ e t p ) = v 1 ( ⊕ t = 1 2 w t τ e t 1 ) ⊕ v 2 ( ⊕ t = 1 2 w t τ e t 2 ) = v 1 ( w 1 τ e 11 ⊕ w 2 τ e 21 ) ⊕ v 2 ( w 1 τ e 12 ⊕ w 2 τ e 22 ) = v 1 { ( min ( 1 , ( w 1 f 11 q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( w 1 ( 1 - g 11 q ) ϑ ) 1 ϑ ) q ) ⊕ ( min ( 1 , ( w 2 f 21 q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( w 2 ( 1 - g 21 q ) ϑ ) 1 ϑ ) q ) } ⊕ v 2 { ( min ( 1 , ( w 1 f 12 q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( w 1 ( 1 - g 12 q ) ϑ ) 1 ϑ ) q ) ⊕ ( min ( 1 , ( w 2 f 22 q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( w 2 ( 1 - g 22 q ) ϑ ) 1 ϑ ) q ) } = v 1 ( min ( 1 , ( ∑ t = 1 2 w t f t 1 q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ t = 1 2 w t ( 1 - g t 1 q ) ϑ ) 1 ϑ ) q ) ⊕ v 2 ( min ( 1 , ( ∑ t = 1 2 w t f t 2 q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ t = 1 2 w t ( 1 - g t 2 q ) ϑ ) 1 ϑ ) q ) = ( min ( 1 , ( v 1 ∑ t = 1 2 w t f t 1 q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( v 1 ∑ t = 1 2 w t ( 1 - g t 1 q ) ϑ ) 1 ϑ ) q ) ⊕ ( min ( 1 , ( v 2 ∑ t = 1 2 w t f t 2 q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( v 2 ∑ t = 1 2 w t ( 1 - g t 2 q ) ϑ ) 1 ϑ ) q ) = ( min ( 1 , ( ∑ p = 1 2 v p ( ∑ t = 1 2 w t f t p q ϑ ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 2 v p ( ∑ t = 1 2 w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) q ) . Hence, result is true for n = 2 , m = 2 .

Suppose Equation 3 is true for n = k 1 , m = k 2 , q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e k 1 k 2 ) = ( min ( 1 , ( ∑ p = 1 k 2 v p ( ∑ t = 1 k 1 w t f t p q ϑ ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 k 2 v p ( ∑ t = 1 k 1 w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) q ) . Now for n = k 1 + 1 , m = k 2 + 1 , q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e k 1 k 2 )

= { ⊕ p = 1 k 2 v p ( ⊕ t = 1 k 1 ( w t τ e t p ) ) } ⊕ v ( k 2 + 1 ) ( w ( k 1 + 1 ) τ e ( k 1 + 1 ) ( k 2 + 1 ) ) = ( min ( 1 , ( ∑ p = 1 k 2 v p ( ∑ t = 1 k 1 w t f t p q ϑ ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 k 2 v p ( ∑ t = 1 k 1 w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) q ) ⊕ v ( k 2 + 1 ) ( w ( k 1 + 1 ) τ e ( k 1 + 1 ) ( k 2 + 1 ) ) = ( min ( 1 , ( ∑ p = 1 k 2 + 1 v p ( ∑ t = 1 k 1 + 1 w t f t p q ϑ ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 k 2 + 1 v p ( ∑ t = 1 k 1 + 1 w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) q ) .

Hence, Equation 3 is true for n = k 1 + 1 , m = k 2 + 1 . Therefore, by mathematical induction the Equation 3 is true, ∀ m , n ≥ 1 .□

Remark 3.1. (a) For q = 1, q - ROFS _f YWA operator becomes intuitionistic fuzzy soft Yager weighted average (IFS _f YWA) operator.

(b) For q = 2, q - ROFS _f YWA operator becomes Pythagorean fuzzy soft Yager weighted average (PFS _f YWA) operator.

(c) For only one parameter, i.e., e₁, q - ROFS _f YWA operator becomes q-rung orthopair fuzzy Yager weighted average (q-ROFYWA) operator.

From Remark 3.1, we conclude that q - ROFS _f YWA operator is generalized case of IFS _f YWA, PFS _f YWA and q-ROFYWA operators.

Example 3.1. Suppose Mr. X wants to purchase a watch from a set of four watches in the domain set U = { x 1 , x 2 , x 3 , x 4 } and let E ˜ = { e 1 , e 2 , e 3 , e 4 } be the set of parameters for the selection of a watch, i.e., e p ( p = 1 , 2 , 3 , 4 ) where e₁ =expensive, e₂ =beautiful, e₃ =synthetic sapphire crystal and e₄ =water resistance. Suppose the WVs for experts x t and parameters e p are w = (0.34, 0.24, 0.20, 0.22)  ^T and v = (0.25, 0.20, 0.30, 0.25)  ^T , respectively. By applying Equation 3, q - ROFS _f YWA (τ _{e 11} , τ _{e 12} , ⋯ , τ _{e 44} ) = ( min ( 1 , ( ∑ p = 1 4 v p ( ∑ t = 1 4 ( w t f t p q ϑ ) ) ) 1 ϑ ) 3 , 1 - min ( 1 , ( ∑ p = 1 4 v p ( ∑ t = 1 4 w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) 3 ) = ( 0.82 , 0.41 ) .

Example 3.2. Suppose Mr. X wants to purchase a car from a set of four cars in the domain set U = { x 1 , x 2 , x 3 , x 4 } and let E ˜ = { e 1 , e 2 , e 3 , e 4 } be the set of parameters for the selection of a car, i.e., e p ( p = 1 , 2 , 3 , 4 ) where e₁ =fuel efficiency, e₂ =good quality, e₃ =safety and e₄ =good warranty. Suppose the WVs for experts x t and parameters e p are w = (0.35, 0.25, 0.15, 0.25)  ^T and v = (0.30, 0.20, 0.20, 0.30)  ^T , respectively. By applying Equation 3, q - ROFS _f YWA (τ _{e 11} , τ _{e 12} , ⋯ , τ _{e 44} ) = ( min ( 1 , ( ∑ p = 1 4 v p ( ∑ t = 1 4 ( w t f t p q ϑ ) ) ) 1 ϑ ) 3 , 1 - min ( 1 , ( ∑ p = 1 4 v p ( ∑ t = 1 4 w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) 3 ) = ( 0.77 , 0.45 ) .

Property 3.1. (Idempotency) If all τ e t p = L e = ( b , c ) ( ∀ t = 1 , 2 , ⋯ , m ; p = 1 , 2 , ⋯ , n ) are identical, then q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) = L e .

Proof. As τ e t p = L e = ( b , c ) ( ∀ t = 1 , 2 , ⋯ , m ; p = 1 , 2 , ⋯ , n ) , then q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m )

= ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) q ) = ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t b q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( 1 - c q ) ϑ ) ) 1 ϑ ) q ) = ( min ( 1 , ( b q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( ( 1 - c q ) ϑ ) 1 ϑ ) q ) = ( min ( 1 , b q ) q , 1 - min ( 1 , ( 1 - c q ) ) q ) = ( b , c ) .

□

Property 3.2. (Boundedness) Let τ e t p = ( f t p , g t p ) ( ∀ t = 1 , 2 , ⋯ , m ; p = 1 , 2 , ⋯ , n ) be a family of q - ROFS _f Ns and τ e t p - = ( min p min t { f t p } , max p max t { g t p } ) and τ e t p + = ( max p max t { f t p } , min p min t { g t p } ) , τ e t p - ≤ q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) ≤ τ e t p + .

Proof. Let τ e t p - = ( min p min t { f t p } , max p max t { g t p } ) and τ e t p + = ( max p max t { f t p } , min p min t { g t p } ) . The inequalities for membership value are min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p - q ϑ ) ) ) 1 ϑ ) q ≤ min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p q ϑ ) ) ) 1 ϑ ) q ≤ min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p + q ϑ ) ) ) 1 ϑ ) q . Similarly, for nonmembership value 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( 1 - g t p + q ) ϑ ) ) 1 ϑ ) q ≤ 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) q ≤ 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( 1 - g t p - q ) ϑ ) ) 1 ϑ ) q . Therefore, τ e t p - ≤ q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) ≤ τ e t p + . □

Property 3.3. (Monotonicity) Let L e t p = ( b t p , c t p ) ( ∀ t = 1 , 2 , ⋯ , m ; p = 1 , 2 , ⋯ , n ) be another collection of q - ROFS _f Ns such that f t p ≤ b t p , g t p ≥ c t p , then q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) ≤ q - ROF S f YWA ( L e 11 , L e 12 , ⋯ , L e n m ) .

Proof. Consider q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) = ( I , J ) and q - ROFS f YWA ( L e 11 , L e 12 , ⋯ , L e n m ) = ( I ′ , J ′ ) . First, we show that I ≤ I′. As f t p ≤ b t p , f t p q ≤ b t p q . Moreover, ∑ t = 1 n ( w t f t p q ϑ ) ≤ ∑ t = 1 n ( w t b t p q ϑ ) ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p q ϑ ) ) ≤ ∑ p = 1 m v p ( ∑ t = 1 n ( w t b t p q ϑ ) ) ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p q ϑ ) ) ) 1 ϑ ≤ ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t b t p q ϑ ) ) ) 1 ϑ min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p q ϑ ) ) ) 1 ϑ ) ≤ min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t b t p q ϑ ) ) ) 1 ϑ ) This implies min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p q ϑ ) ) ) 1 ϑ ) q ≤ min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t b t p q ϑ ) ) ) 1 ϑ ) q . Therefore, I ≤ I′. Similarly we can show that J ≥ J′. Hence, q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) ≤ q - ROFS f YWA ( L e 11 , L e 12 , ⋯ , L e n m ) . □ Property 3.4. (Shift Invariance) If L e = ( b , c ) is another q - ROFS _f N, then q - ROFS f YWA ( τ e 11 ⊕ L e , τ e 12 ⊕ L e , ⋯ , τ e n m ⊕ L e ) = q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) ⊕ L e .

Proof. Since L e = ( b , c ) and τ e t p = ( f t p , g t p ) are q - ROFS _f Ns, so

τ e 11 ⊕ L e = ( min ( 1 , ( f 11 q ϑ + b q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( ( 1 - g 11 q ) ϑ + ( 1 - c q ) ϑ ) 1 ϑ ) q ) .

Therefore, q - ROFS f YWA ( τ e 11 ⊕ L e , τ e 12 ⊕ L e , ⋯ , τ e n m ⊕ L e ) = ⊕ p = 1 m v p ( ⊕ t = 1 n w t ( τ e t p ⊕ L e ) ) = ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( f t p q ϑ + p q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( ( 1 - g t p q ) ϑ + ( 1 - c q ) ϑ ) ) ) 1 ϑ q ) ) .(4) q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) ⊕ L e = ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) q ) ⊕ ( b , c ) = ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) q ) ⊕ ( min ( 1 , b q ) q , 1 - min ( 1 , ( 1 - c q ) ) q ) = ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) q ) ⊕ ( min ( 1 , ( b q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( ( 1 - c q ) ϑ ) 1 ϑ ) q ) = ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) q ) ⊕ ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t b q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( 1 - c q ) ϑ ) ) 1 ϑ ) q ) = ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( f t p q ϑ + p q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( ( 1 - g t p q ) ϑ + ( 1 - c q ) ϑ ) ) ) 1 ϑ q ) ) .(5)

From Equations 30 and 31, q - ROFS f YWA ( τ e 11 ⊕ L e , τ e 12 ⊕ L e , ⋯ , τ e n m ⊕ L e ) = q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) ⊕ L e . □ Property 3.5. (Homogeneity) Let λ ≥ 0 then q - ROFS f YWA ( λ τ e 11 , λ τ e 12 , ⋯ , λ τ e n m ) = λ q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) .

Proof. As λ τ e t p = ( min ( 1 , ( λ f t p q ϑ ) 1 ϑ ) q , 1 - min ( 1 , ( λ ( 1 - g t p q ) ϑ ) 1 ϑ ) q ) . Now q - ROFS f YWA ( λ τ e 11 , λ τ e 12 , ⋯ , λ τ e n m ) = ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( λ f t p q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( λ ( 1 - g t p q ) ϑ ) ) ) 1 ϑ ) q ) .(6) and q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) = ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n ( w t f t p q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( 1 - g t p q ) ϑ ) ) 1 ϑ ) q ) . λ q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) = ( min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( λ f t p q ϑ ) ) ) 1 ϑ ) q , 1 - min ( 1 , ( ∑ p = 1 m v p ( ∑ t = 1 n w t ( λ ( 1 - g t p q ) ϑ ) ) ) 1 ϑ ) q ) .(7)

From Equations 32 and 33, q - ROFS f YWA ( λ τ e 11 , λ τ e 12 , ⋯ , λ τ e n m ) = λ q - ROFS f YWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) . □

q - ROFS _f YWA operator only weighted the q - ROFS _f N′s values. But q - ROFS _f YOWA operator weights the ordered positions via scoring the q - ROFS _f values rather than weighting the q - ROFS _f values themselves. In this section, we develop q - ROFS _f YOWA operator and related properties.

Definition 4.1. Let τ e t p = ( f t p , g t p ) ( t = 1 , 2 , ⋯ , n ; p = 1 , 2 , ⋯ , m ) be a family of q - ROFS _f Ns and w t , v p are the WVs for the experts x t ′ s and parameters e p ′ s , respectively, satisfying w t > 0 , v p > 0 with ∑ t = 1 n w t = 1 and ∑ p = 1 m v t = 1 . Then q - ROFS _f YOWA operator is a function q - ROFS _f YOWA: M n → M q - ROFS f YOWA ( τ e 11 , τ e 12 , ⋯ , τ e n m ) = ⊕ p = 1 m v p ( ⊕ t = 1 n ( w t τ σ e t p ) ) .